High Rate Efficient Local List Decoding from HDX

We construct the first (locally computable, approximately) locally list decodable codes with rate, efficiency, and error tolerance approaching the information theoretic limit, a core regime of interest for the complexity theoretic task of hardness amplification. Our algorithms run in polylogarithmic time and sub-logarithmic depth, which together with classic constructions in the unique decoding (low-noise) regime leads to the resolution of several long-standing problems in coding and complexity theory: 1. Near-optimally input-preserving hardness amplification (and corresponding fast PRGs) 2. Constant rate codes with $\log(N)$-depth list decoding (RNC$^1$) 3. Complexity-preserving distance amplification Our codes are built on the powerful theory of (local-spectral) high dimensional expanders (HDX). At a technical level, we make two key contributions. First, we introduce a new framework for ($\mathrm{polylog(N)}$-round) belief propagation on HDX that leverages a mix of local correction and global expansion to control error build-up while maintaining high rate. Second, we introduce the notion of strongly explicit local routing on HDX, local algorithms that given any two target vertices, output a random path between them in only polylogarithmic time (and, preferably, sub-logarithmic depth). Constructing such schemes on certain coset HDX allows us to instantiate our otherwise combinatorial framework in polylogarithmic time and low depth, completing the result.

💡 Research Summary

The paper presents the first family of (locally computable, approximate) locally list‑decodable codes (aLLDCs) that simultaneously achieve near‑optimal rate, efficiency, and error tolerance, approaching the information‑theoretic limit. The constructions are based on high‑dimensional expanders (HDX) and introduce two major technical innovations. First, the authors develop a new belief‑propagation framework that runs for polylogarithmic rounds on HDX. By interleaving local correction steps with global expansion, the algorithm controls error accumulation while preserving a constant encoding redundancy, thereby maintaining a rate close to 1 − H(½ − ε). Second, they define strongly explicit local routing on HDX: given any two vertices, a random short path can be produced in polylogarithmic time and sub‑logarithmic depth, even when a constant fraction of vertices are faulty.

Using these tools, three regimes of aLLDCs are constructed. (1) Sub‑polynomial‑rate codes are built from a subspace set system together with low‑congestion routing, yielding polylog(N) query complexity for list recovery. (2) Polylog‑rate codes are obtained from the KMS (coset) complex, a specific HDX whose links admit deterministic routing that is transformed into low‑congestion routing. This yields codes with rate arbitrarily close to 1, polylog(N) query complexity, and polylog(N) rounds of belief propagation. (3) Constant‑rate codes are achieved by composing an inner decoder (based on the KMS set system) that produces a 99 % accurate list, with an outer decoder that prunes the list into a well‑separated form. The combined decoder runs in O(log N) depth, i.e., in RNC¹, providing the first log‑depth list‑decoders for the high‑noise regime (error up to ½ − ε).

The paper also proves matching information‑theoretic lower bounds, showing that any code with error tolerance ½ − ε must satisfy rate ≤ 1 − H(½ − ε) + o(1), and the presented constructions meet this bound up to lower‑order terms.

Beyond coding theory, the authors demonstrate three major applications. (i) Input‑preserving hardness amplification: encoding a weakly hard function f with an aLLDC yields a function Enc(f) that is ½ + ε hard against the same class of algorithms, while the encoder remains locally computable. (ii) Complexity‑preserving distance amplification: composing a base code with an aLLDC amplifies its error‑correction capability without increasing encoding or decoding circuit complexity. (iii) Efficient pseudorandom generators: the hardness‑amplified functions lead to near‑linear‑time PRGs with optimal seed length.

Overall, the work resolves several long‑standing open problems: it provides the first high‑rate aLLDCs, achieves log‑depth parallel list decoding in the high‑noise regime, and supplies tools for hardness amplification and distance amplification that preserve computational complexity. The combination of belief propagation on HDX and strongly explicit local routing constitutes a powerful new algorithmic paradigm for high‑dimensional combinatorial structures.